Abstract
An approach for minimizing risk through diversification of foreign exchange reserves and sovereign borrowings is proposed for central banks of small open economies. This approach--developed in a simple 2-period, 3-country framework--differs from past work in that the elements of exchange and price risk associated with trade and payments are considered in the portfolio allocation problem. The analysis shows that the net level of reserves and the primary transactions balance affect the optimal portfolio leading to deviations from the optimal allocation prescribed by the classical portfolio model. In addition, this result has implications for the currency composition of exchange market intervention transactions.
I. Introduction
The substantial fluctuations in exchange rates in recent years have had a significant impact on the purchasing power of international reserves. One approach to reducing the risk of a loss in purchasing power resulting from a depreciation in one of the major reserve currencies is to diversify holdings of foreign exchange reserves across a number of currencies. Such diversification has indeed taken place over the last 20 years and there is a fairly extensive literature describing this development. Some of the contributions to this literature have aimed at identifying those variables that can account for actual holdings in different currencies and have focused on explaining the observed diversification behavior of central banks and monetary authorities (see, for example, Heller and Knight (1978), Dooley, et al. (1989), and Ben-Bassat (1980, 1984). Other authors have been more concerned with analyzing the appropriate approach that should be taken by central banks in diversifying their foreign exchange reserves, e.g., Healy (1981) and Lehmussaari (1987).
This paper falls more in the latter category. It does not attempt to explain the actual diversification of reserve portfolios across currencies by central banks. Rather, it describes a framework for the management of reserves by focusing on the basic function of reserves to meet the excess demand for foreign exchange. Reserves in this analysis are viewed as a measure of aggregate wealth or purchasing power of the country, and minimizing the risk associated with this purchasing power is assumed to be a plausible objective of the central bank. This portfolio approach for the minimization of risk, extends previous analysis by taking account of a country’s payments and receipts for goods and services in the currencies of invoicing. This approach is therefore broader in that it takes account of a wider array of foreign payments than the classical portfolio model, where the portfolio choice depends only on invested assets, and excludes uncertainties associated with other payments and receipts, (see, for example, Sharpe (1981), Stulz (1981), Adler & Dumas (1983), de Macedo (1983), Claessens (1988), Madieros and Nocera (1988). Thus this approach can be viewed as a more general portfolio approach that relates the interest earned on a country’s foreign currency reserves to the trade and invisible components of the balance of payments, thus integrating the external current account into the portfolio problem, and the related implications of the diversification strategy pursued by a country. Finally the paper identifies a relationship between the level of net reserves and the optimal currency allocation, which has been possible because of this generalization and consequently has not been established in earlier portfolio models for reserve management. The objective for the central bank of minimization of risk related to the purchasing power of the country is maintained throughout in the interest of clarity and separates the contribution of this paper from that of the classical mean- variance model. This treatment is, however, consistent with and may be extended to include conditional minimization of risk given some level of expected return as in the classical mean-variance theory.
The framework developed here has useful implications for the composition of sovereign debt as well as that of central bank reserves. This stems from the fact that sovereign debt is a liability and may consequently be regarded as negative reserves. Consequently, the currency composition of either the reserves or sovereign debt would affect the currency and interest rate exposure of the country. Private borrowings are exogenous in the model and are assumed to be uncontrolled 1/ by the central bank and the government.
The consideration of the current account imbalance in this portfolio model results in some interesting deviations from both the classical model, and also from a portfolio strategy of simply hedging payments and receipts with investments in the currencies of these payments and receipts. For example, the prescribed portfolio here could deviate from the simple hedge strategy by including a holding in a currency where there is no current account imbalance. On the other hand, examples of deviations from the classical portfolio model include the importance of the level of Reserves and the effects of the volatility of prices on the optimal currency allocation of reserves.
The paper is divided into eight sections. Section II describes the role of international reserves in meeting the excess demand for foreign exchange at the exchange rate desired by the monetary authorities. Against this background, it discusses a plausible strategy for the authorities to follow in diversifying its foreign exchange reserves, namely, minimize the variance of changes in its net reserves. It is argued here that the basket of foreign purchases that is relevant for a central bank is not restricted to a consumption bundle made up of, e.g., merchandise imports, but rather comprises the bundle of foreign payments regularly made by the country to foreign residents. In this context the minimization of the variance of changes in the level of real reserves and sovereign borrowings may be viewed as smoothing the fluctuation of aggregate purchasing power of the country.
The next section then presents the portfolio model that allocates reserve assets in the appropriate proportions, such that the resulting diversification of reserve assets takes account of the structure of a country’s international transactions. Using a simple two-asset, three-country framework, the model provides a method for incorporating the co-variance between the return on reserve assets and other current account transactions in minimizing the variance of the changes in the value of reserves. The shares of reserve assets that minimize this variance are shown to depend on the relative magnitudes of the variance of the exchange rates and the pattern of the country’s foreign receipts and payments in terms of invoicing currencies. The relationship between the reserve level and the optimal portfolio composition is then established.
Although the discussion in the sections that follow focus on the problem of diversification of central bank reserves, there are corresponding implications for the diversification of sovereign borrowings. Some such considerations are addressed in section IV.
Section V presents a numerical example that contrasts the results of a conventional portfolio allocation model constructed to minimize variance with the more general model proposed here. In addition, the effects of changes in the reserve level to the optimal currency composition is examined. The following section identifies some significant implications for the use of reserves - including, most importantly, exchange market intervention.
Section VII of the paper broadens the scope of the discussion to include a number of considerations influencing the choice of currencies in which to borrow or invest reserve assets. These include the exchange rate objectives of the country, the intervention currency used to achieve these objectives, the liquidity of the instruments in which reserves are invested, and transactions and other costs that have a bearing on the management of reserves. It is argued that while these additional factors obviously need to be taken into account in arriving at the allocation across currencies appropriate for a particular country, they do not vitiate the approach for the diversification of reserves and sovereign borrowings described in the earlier sections of the paper. The conclusions follow.
II. The Relationship of Reserves to the Structure of a Country’s International Transaction
International reserves serve a number of functions for central banks. Among these are a store of national wealth to be used for emergency imports, and as implicit collateral to lower the cost of foreign borrowing. Further, the use of reserves in foreign exchange market intervention to finance the gap between the total supply of foreign exchange forthcoming at the exchange rate desired by the authorities, and the quantity demanded at that same exchange rate is assumed to be exogenous to the model. This exchange market intervention function of reserves is related to the exchange rate objectives of the country. In particular, the amount of reserves held will tend to be influenced by the desired degree of flexibility in the exchange rate of the domestic currency in terms of one or more reserve currencies. This aspect of the reserve holdings, i.e., the optimal level of reserves, is not considered here. Rather, the optimal composition is determined, given that the central bank has chosen its preferred level of reserves based on its desired level of flexibility in its intervention strategy.
In the analysis presented, it is shown that the currency composition of transactions as in exchange market intervention are influenced by the optimal currency allocation of reserves, it is assumed that transactions costs are sufficiently low that they are not a major determinant of the desired composition of reserves. However, if these costs are significant, they could affect the choice of the vehicle for intervention. As has been emphasized by Dooley, et al (1989), the extent of this influence depends on the magnitude of the transaction costs. This factor, as well as other institutional aspects of reserve holding, are dealt with in Section VII.
It needs to be recognized at the outset that the exchange rate objective of a country may well influence the currency composition of its reserves to the extent that exchange market intervention influences the exchange rates of the major industrial countries. In particular, as the G-7 have particular exchange rate objectives for their currencies vis-à-vis each other, these objectives would no doubt take precedence over any diversification considerations. It is, therefore, unlikely that the approaches to currency diversification described in the literature, as well as that discussed below, would be applicable to the major industrial countries. Rather, they are particularly relevant for the smaller industrial countries and developing countries that are price takers with regard to the exchange rates between the currencies of the G-7.
The starting point of the method used here for deriving a diversified portfolio of international currency reserves is the recognition that these foreign assets are used to finance the net excess demand for payments abroad generated by the private and government sectors (other than the central bank or monetary authority)2/. As most small economies (many of them developing countries) do not have flexibility in financing their net deficits with their capital accounts, through appropriate monetary policy and resulting changes in domestic interest rates, such financing will depend on available reserves and sovereign debt. Thus, this role of reserves may be viewed as that of meeting the net claims on the country for goods and services. In this context, changes in reserves may be considered to represent changes in aggregate wealth of the country. Although the framework assumes that the composition of sovereign debt is controlled by the authorities, the results proposed will hold under the alternative scenario where the denomination of sovereign debt is exogenously-determined by available credit, and the composition of central bank reserves is controlled to meet the proposed objective, i.e., the net reserve composition is controlled. Although formally this simple two period framework includes only the implicit assumption that the central bank meets the demand for foreign exchange from its reserves to cover net external payments, in principle 3/ the objective will not be restrictive for a broader intervention “policy” framework where policy related intervention is infrequent in relation to that for matching imbalances in the primary account. In this context the control of exchange rates by the authorities for policy purposes is of primary concern to the government and the central bank. Thus changes in exchange rates resulting from such voluntary policy intervention cannot be considered to be an element of risk faced by the central bank. Consequently, any reserve management program should be subordinated to these policy objectives. Therefore, consideration of such periods of policy related intervention to voluntarily change the exchange rate, in this framework, would not be consistent with the broader objectives of the central bank. In this context therefore, the overall exchange rate or exchange rate management system adopted by the central bank is not of interest here. Whatever the approach to exchange rate management, fluctuations in a country’s international payments and receipts result in net demands for foreign exchange that are met by the central bank at times when there is no active intervention to change the exchange rate. It therefore follows in this context that if attention is focused on fluctuations in the value of exchange reserves, then it is necessary to look at all the transactions that generate the demand for foreign currency that is supplied by the monetary authority.
A simplified balance of payments identity is the starting point for examining this function of reserves:
where:
| ei | = | units of numeraire currency per unit of currency i |
| PEi | = | price vector of exports in currency i |
| Ei | = | quantity vector of exports priced in currency i |
| rio | = | equivalent domestic interest rate on instrument denominated in currency i (including the effects of exchange movements) |
| Ri | = | value of reserves held in currency i |
| PIi | = | price of imports in currency i |
| Ii | = | quantity of imports that are priced in currency i |
| ei | = | units of numeraire currency per unit of currency i |
| PEi | = | price vector of exports in currency i |
| Ei | = | quantity vector of exports priced in currency i |
| rio | = | equivalent domestic interest rate on instrument denominated in currency i (including the effects of exchange movements) |
| Ri | = | value of reserves held in currency i |
| PIi | = | price of imports in currency i |
| Ii | = | quantity of imports that are priced in currency i |
Equation (1) is a balance of payments identity expressed in terms of the domestic currency. The changes are expressed over suitable period of time e.g. a day or year. There are assumed to be two components of receipts--merchandise exports and earnings on foreign exchange reserves—and one type of payment—merchandise imports. Thus private capital flows, services, and transfers are excluded in this simple version of the identity, and hence any positive (negative) difference between receipts from merchandise exports and interest earnings on reserves and payments for imports is added to (subtracted from) reserves.
In equation (1) the valuation of receipts and payments is currency 4/, not country specific. The prices denoted are those that are determined in the underlying markets for the goods in question. In many situations these would be the domestic prices for exports. However, when a country is a price taker on world markets, then the relevant price is that determined in those markets, e.g., the dollar price of wheat or petroleum. The same holds true for many import prices of most countries. Finally, the currency denomination of the reserve asset may differ from the currency of the country of residence of the issuer of the liability, as in the case of eurocurrency deposits.
Equation (1) can be rearranged to combine all transactions involving the same currency to give the following:
From equation (2) it can be seen that investing some foreign exchange reserves in instruments denominated in the same currencies as imports will protect the country to some extent from exchange rate movements that increase import costs; i.e., an exchange rate change that increases import payments in foreign currency (PIi,Ii) can in part be matched by increases in service receipts (roiRi). Thus if a portion of the stock of reserves is allocated to the same currency as a component of imports, the increased cost of imports generated by depreciation of the exchange rate (a rise in the value of e) is at least partially offset by the capital gain, as measured by the increase in the value of reserves in currency i in terms of numeraire. This provides the basic rationale for distributing foreign exchange reserves in relation to the underlying currency composition of imports, which is the starting point of most approaches to currency diversification.
The gains from general diversification across currencies in reducing the variance of a portfolio of foreign currency assets is well known. The importance of taking account of the currency composition of a country’s imports has also been analyzed in terms of achieving the objective of reducing the variability of the real value of a country’s international reserves. The approach suggested here is more general in that it takes account of the possibility of offsetting movements in the components of a country’s international transactions. In particular, as shown in equation (2), changes in the value of exports can offset, or magnify, the impact of variations in payments for imports on the level of reserves. Therefore in devising a portfolio strategy that aims at minimizing the volatility of the real value of a country’s reserves, the covariation in the major components of all of a country’s international transactions need to be integrated with the covariation of invested reserves in foreign currencies.
More generally, the argument above needs to be extended to the entire range of a country’s international transactions and currencies for investment. Under this approach, all current and capital account transactions would be grouped by currency, or more specifically, by the excess demand for a currency. This more general formulation would include the share in total foreign exchange reserves that is allocated to a particular currency, (αi). Here investment is possible even in currencies where there is no trade imbalance. This generalized approach to reserve diversification can be represented by the following equation:
where:
| αi | = | Ri/R |
| Di | = | total demand for foreign currency i |
| Si | = | total supply of foreign currency i exclusive of interest earning on reserves |
| αi | = | Ri/R |
| Di | = | total demand for foreign currency i |
| Si | = | total supply of foreign currency i exclusive of interest earning on reserves |
The model assumes that the objective 5/ is to minimize the risk of changes in the real value of reserves. Minimizing the uncertainty in changes in reserves may be viewed as smoothing changes in the aggregate wealth or purchasing power of the country. Risk here is measured as being proportional to the variance of changes in the value of reserves and stems from the stochastic nature of exchange rates, foreign interest rates (and the deflator if changes in real reserves are considered). From the above equation it may be seen that this will be achieved by the program:
This minimization problem is subject to the constraint that Σ αi = 1.
In addition, if borrowing by the monetary authority is ruled out6/, one can impose the constraint that αi ≥ 0. Sovereign debt however, implies that αi ‹ 0 for some i and possibly R ‹ 0.
This method for determining the foreign currency diversification of reserves is more general than standard approaches because (1) it takes account of more transactions than only payments in foreign currency for merchandise imports, and (2) it recognizes the stochastic nature of these other transactions. It involves looking at returns on foreign currency reserves as an integral part of a country’s international transactions, with the currency composition designed to hedge the particular exposures in individual currencies that arise in the course of a country’s trade and financial activity.
III. The Reserve Portfolio Model 7/
The model used here to illustrate the central management of foreign reserves is based on a two-period model. In this model it is assumed that the central bank has the objective of minimizing the variance of changes in the real value 8/ of net reserves, which in this context is a measure of the nations endowment.
The case of two foreign countries with trade and finance relationships to the home country is considered in the interest of simplicity. As before let the exchange rate of the two countries in units of the domestic currency be e1 and e2. Let there be two goods traded with each country an export and an import, with prices PE1, PI1, PE2, and PI2 respectively. The interest rates on treasury instruments which are assumed representative of returns in countries 1 and 2 are r1 and r2. Also, let the total currently outstanding private debt due to the two countries be give by D1 and D2 respectively. Finally, let the export volume of the goods to the two countries be give by E1 and E2 and the corresponding import volumes be I1 and I2. The domestic currency value of merchandise imports and interest payments on foreign debt is given by:
As there are only two foreign countries in the model, the central bank holds its reserves in these currencies in the amounts, R1 and R2. R* denotes the total real value of reserves i.e., deflated by the import basket of goods and services P0I9/.
The objective of the model is to find the optimal (variance minimizing) shares of reserves that are invested in the two foreign currencies. Let these shares be denoted by α1 and α2. Consider the return obtained by the central bank in the foreign instrument. The return will be based on the current exchange rate, in converting to the foreign currency and then the rate of return in the foreign bond market and finally the rate of exchange obtainable in converting back to the domestic currency. This return will be:
Here the returns are the total returns 10/ and the - indicates that the variable is a random variable at time ‘t’ and (r0(t)) denotes the returns in domestic terms. Consequently, the total reserve position invested in each currency at the end of the “t”th period due to income and capital gains, is given by,
In addition, the country’s imports, exports and debt service payments will change the level of the reserves if there is no concurrent policy related intervention. This change in reserves will reflect the behavior of the “primary” balance of payments, i.e., all international transactions except interest earnings (payments) on reserves (debt) by the central bank.
The changes in reserves generated by these transactions are given by,
Thus, the total reserves at the end of the ‘t’th period will be,
If it is assumed that the quantities of the foreign payments, i.e., goods and services, is fixed for the time period considered then the reserves in terms of the basket of foreign payments may be given by,
Separating the reserves R*(t+1) into its components reflecting changes in the primary balance of payments (RT) and returns on investment (RI), the following relationships are established.
Further let
and
Now as R*(t+1) from equation (8) is the real reserves at the end of the “t”th period, the objective would be to minimize its variance by changing αi. To achieve this it is necessary to identify the variance and the covariances of its components. These components depend on the balance of payments, in addition to the reserve level at time t+1, the interest rates, exchange rates, and the price index at time t and may be represented by a vector,
The covariance of this vector may be represented by the matrix V (4x4), and expected value may be given by a. In the following analysis it is assumed that the real reserve distribution is a member of distributions that may be represented by two parameters; the mean and the variance, e.g., the Normal distribution.
As the objective is to minimize the components of the variance contributed by these elements, the optimization program to minimize the variance of reserves is given by:
such that α1 + α2 = 1
This program finds the weights α1 and α2 such that the variance of the portfolio is minimized and the weights of the elements of the portfolio add up to one.
Forming the Lagrangian,
Taking first order conditions and solving, the composition of reserves is given by α*1 (the fraction α*2 is simply 1-α*1).
These portfolio shares may also be expressed in terms of the variances and covariances of the interest and exchange rates and those of the prices (see Appendix).
Now consider a simplification to aid exposition without loss of the basic intuition in this model. If prices were non stochastic and the net private debt position was zero, then the following simplifications are possible. (see the Appendix for the general case).
Here γi = EiPEit eit - Ii PIit eit
and Vij is the covariance between i and j, and Vi is the variance of i,
Substituting in [12] we have,
This is the solution for the composition of reserves for the special case with deterministic prices. Specific properties of this solution are now examined.
Consider the effect of the variation of the level of net reserves with the optimal allocation. It is seen from the above expression that the currency allocation is inversely proportional to the level of reserves or sovereign debt. It may be inferred from this that each country has an optimal currency mix for debt or reserves depending on its desired level of debt or reserves. Now to examine the rate of decrease in the portfolio weights with the increase in reserves, differentiate the above solution with regard to the reserves R.
It is readily seen that the currency allocation weights become increasingly sensitive to changes in reserves as the level of reserves fall. A useful inference is that countries that have low net reserve levels relative to primary surpluses or deficits need to make larger adjustments to their currency allocations as reserve levels change. On the other hand a large holding of reserves or sovereign debt relative to the primary imbalance reduces the need for significant changes in the composition of reserves (or debt). This reflects the compensating effect of the investment portfolio on the imbalance in the primary account. This compensation effect stems from the fact that when the level of reserves relative to the imbalance is small, the impact on investment income of any changes to the composition of reserves will be limited. Thus a larger adjustment in composition is necessary to arrive at the optimal mix of investments, to balance the uncertainty from the primary account imbalance.
It is also interesting to note that the currency allocation problem will be independent of the level of reserves, when the primary payments balance in each currency bears a particular relationship to covariances of interest rates and exchange rates. This condition is as follows:
This relationship represents the compensation effect between investments and the primary balance in each currency, when the covariance of these currencies bear a particular relationship to the imbalance in the primary account in the two currencies.
Now consider the special case when the primary imbalance is only in currency 1. Then, α2 = 0. Here α1 is given by,
and is not in general equal to one, as would be expected for a simple hedge where available reserves are held in currency 1 to compensate for transactions. Thus, the optimum portfolio includes some investment in currency 2 to lower risk further than if only currency one was used for a hedge position.
In the special case when the net reserve position is zero, there is no prescription on investment policy given by the model, and the related “a” (i.e., buying long in some currencies and selling short in others).
Consider the special case of equation [22] when γ1=γ2=0. In this case the solution is that of the classical model for variance minimization, as the optimal portfolio allocations are independent of the primary account and consequently the level of reserves won’t matter, whatever the correlation structure of exchange rates interest rates and prices.
Now consider the case when γ1=γ2≠0. Here the condition for independence of the portfolio allocation and reserve levels reduces to,
This may be interpreted as a balancing influence among similar covariance terms and in turn requires divergences from symmetries in these terms to compensate each other. For example, in the unlikely event of the variances of exchange rates being equal, the covariances of exchange rates with interest rates in the same currency being equal and covariance of exchange rates with interest rates in the other currency being also equal, the portfolio allocation will not depend on reserve levels.
Now further insights can be gained with additional simplification of the solution in [20]. We proceed as follows. Assume that only the exchange rates are stochastic. Then,
In the special case where exchange rates are uncorrelated, this simplifies to,
It may be seen from the above that in this special case, the larger the surplus in the primary account in any currency the smaller the proportion of the net reserves held in that currency, and the larger the investment in the other currency. This relationship is due to the additional diversification necessary away from the currency with the larger imbalance in the primary account, and the higher related variance, to maintain the minimum variance portfolio. This result, of course, holds for negative balances when the result is intuitively more appealing, as the negative balance may be viewed as needing an investment in that currency to offset risk. It is however not possible in general to perfectly offset the risk associated with the trade imbalance as the exogenous net reserve level acts as a constraint.
Finally, if the value of primary transactions net to zero, i.e., γi=0,
As may be seen in this case, the portfolio proportions are proportional to the variance of the other currency rate of exchange. This corresponds to the well known property of minimum variance portfolio selection in financial economics. It is interesting to note that the level of net reserves do not matter in this special case when trade payments net to zero. This is an important distinction between the general model presented in this paper and that often used with the above simplifying assumption.
IV. Implications for the Currency Composition of Sovereign Debt
Now consider countries that have a net liability position. The aggregate wealth of these countries will be negative to reflect these liabilities. The goal of minimizing the variance of changes in net reserves will thus effectively minimize the variance or risk in the changes in aggregate wealth or purchasing power of such countries. It will be demonstrated in the following section that the optimal composition of sovereign debt for smoothing aggregate wealth or the purchasing power of the country depends on the currency composition of the primary balance. Consequently, each country has an optimal currency composition for borrowing. Most notably, it may be seen from the example in the next section that the optimal currency composition is sensitive to levels of borrowing relative to the primary imbalance.
This result has implications for fixed pool lending such as with the SDR of the IMF. 11/ Ideally a facility with a variable mix of currencies to suit the primary imbalance of the country would be most attractive. However, transactions costs especially for commercial debt and the availability of a swap market could somewhat mitigate the disadvantages of fixed pool borrowings. The exact magnitude of the deviations in the optimal composition of borrowings from that prescribed by the classical model needs to be established empirically.
V. An Illustrative Numerical Example
The following example illustrates many of the characteristics of the portfolio model presented here. Comparison with the classical portfolio model yields interesting divergences, in the optimal allocation of currency reserves. Again, in the interest of clarity of exposition, the level of complexity is reduced by keeping prices non-stochastic.
Consider a country with two trading partners the United States and Germany and with the facility to invest in the currencies of these countries dollars and marks. We assume for the purpose of this example that the prices of traded goods and the price index for the import basket are deterministic. The central bank of the country has to allocate its reserves under different scenarios. These include most importantly, the level of reserves, and characteristics of the random nature of exchange rates and interest rates in the two foreign countries. We assume for the base case that the interest rates and exchange rates are all correlated (with a correlation coefficient 0.5) and that the standard deviations on the interest rates and the returns on the exchange 12/ rate are 0.005. These would be in a range similar to a broad group of countries and the results would be significant for the practice of reserve management and optimal sovereign borrowings for such countries.
In the interest of studying the effect of reserve level on optimal composition, the country is initially assumed to have a varying level of net reserves, R, valued in the domestic (numeraire) currency. Let the primary surplus excluding interest on debt for the two countries be given by γ1 and γ2 as in the last section. Then equation [20] yields the relationship between the level of reserves (or sovereign borrowings) and the optimal composition of reserves as illustrated in chart 1. As there are only two foreign currencies the portfolio holdings are α1 and (1-α1). It should be noted that there can be short sales of currencies, i.e., borrowings by the central bank. If the country is a net borrower, then the implications here are that it borrow additional funds in the currency sold short, to invest in the other currency. In practice however, credit rating constraints will restrict the gross short positions that countries can take. In this example the level of reserves in relation to the primary surplus or deficit is chosen to be in a range of up to about 100 days net imports or exports. (Please see chart 1.)
Currency Composition of Reserves Variation with Reserves or Sovereign Debt
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
In the base case where there is a primary surplus in both currencies of 1 domestic unit, the optimal holding of reserves is not affected by the level of reserves. This result arises from the symmetry in the model by construction of the covariances among the random elements and the primary surpluses γ1 and γ2. This result follows as equation [24] is satisfied, and the result is consequently consistent with the classical portfolio theory where the level of reserves is not related to the optimal allocation of currencies. This result also follows in the case where the primary balance is symmetric between the currencies. The situation is different, however, when there is asymmetry in the primary transactions balances with the two countries. Here it is seen that if there is a greater primary surplus with Germany, the optimal allocation to dollars rises sharply as the level of reserves get smaller, and conversely when the primary surplus with the United States is greater, then the allocation to the dollar falls to increasing negative levels (short selling) with decreases in reserves. It is also of interest to note that the discontinuity in the prescribed value of “α1” as the level of net reserves crosses zero in the diagram is due to the change in sign of the terms with R in equation [20].
Now consider the reserve level fixed at 30 domestic units. Consider primary surpluses in (dollars, marks) of (2,1), (1,2), (-4,6) and (6,-4). These cases represent asymmetries of different levels between the surpluses/deficits in the two countries, which may be compared with the symmetric surpluses of (1,1) and (-1,-1) (which yield results exactly the same as the conventional portfolio problem).
It is instructive to study the effects of changes in the standard deviation of interest rates and exchange rates to examine the effects of the primary balances, which play no role in the conventional portfolio theory. First consider the variation in the standard deviation of the interest rate in the dollar investment. The results are shown in chart 2 and 3.
Currency Composition of Reserves Variation with Standard Deviation of r1
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
Currency Composition of Reserves Variation with Standard Deviation of r2
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
From chart 2 it may be seen that when the dollar interest rate has a low standard deviation, in all cases - whether the surplus with the United States is greater or smaller, the optimal allocation to dollars is large. Naturally the optimal allocation is greater when the surplus with Germany rises with an increased need to diversify risk with dollar allocations. On the other hand, when the standard deviation of the dollar interest rate is large there is little point in investing in dollar assets as it cannot contribute towards decreasing the overall variance. As indicated above, the optimal allocation to dollars is small (and possibly negative) in this case. In situations where there is a larger surplus with the United States, the compensation effect decreases the holding of the dollar investment. When the variance of the asset increases this deviation in the investment in the dollar diminishes as dollar investments become more risky and dominate the compensation effect. As indicated above, as a result the deviations from the classical model decrease as it becomes increasingly difficult to diversify risk. Also, both the total surplus as well as the surplus with the United States, matter in the optimal allocation as may be seen in equation [20], and in chart 2. As the total surplus rises there is a steeper fall in the dollar investment with rising dollar interest rate variance. Chart 3 illustrates similar conclusions with the variation of the standard deviations of the interest rate in Germany.
The effect of changes in the standard deviation of exchange rates are different to those above in that exchange rates affect both the investments and the transactions in dollars and marks. As may be seen below the adjustment in the investment from that prescribed in the classical model in the dollar (and mark) (same as the case with no net primary surplus) rises as the magnitude of the surplus/deficit in dollar rises. As a result although the increase in the variance of the dollar exchange rate, decreases the investment in dollar securities, the effect of the higher variance in the dollar surplus is also a factor in the portfolio allocation. As may be seen in chart 4 and 5, this results in a steeper fall in the dollar investment with rising dollar variance for the case of a higher dollar surplus, and a shallower fall when the mark surplus is greater. Further, a rise in the total surplus results in an increasing downward bias in the investment in dollars as its exchange rate variance rises.
Currency Composition of Reserves Variation with standard deviation of e1
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
Currency Composition of Reserves Variation with Standard deviation of e2
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
Now consider the effects of the correlation of the stochastic elements on the generalized model to examine the divergence from the conventional portfolio model. The following illustrate the effects of the correlation of interest rates and the exchange rates for different levels of primary-surplus with the United States and Germany. First consider the correlation between the exchange rates as illustrated in chart 6.
Currency Composition of Reserves Variation with Correlation (e1,e2)
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
As the exchange rate affects both the investments and the primary balance, the compensating effect of balances in the currency with the lower primary surplus are reduced as the exchange correlation rises. This is because the compensating effect becomes less effective, however, there is an added benefit in holding a diversified (long in both currencies) portfolio, in reducing overall variance. Thus the optimal currency holdings converge as the correlation approaches unity. As may be seen in the chart the currency allocations are unchanged when the primary balance is symmetric between currencies. Now consider a higher variance in the dollar exchange rate as in chart 7. Here as the correlation reaches unity, the optimal portfolio shifts towards the low variance currency - the mark. The transfer to marks is greater when the primary surplus is larger especially when it is predominantly with Germany.
Currency Composition of Reserves Variation with Correlation (e1,e2)
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
The effect of correlations between interest rates however affect only the investments and not the primary balances. Consequently greater compensation can be realized by increasing the reserve allocation to the currency with a lower surplus, with the benefit of returns in the two currencies moving in concert as the correlation increases. See chart 8.
Currency Composition of Reserves Variation with correlation (r1,r2)
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
Now consider the correlation of the exchange rate with the interest rate in each currency. Here there is an interesting instability in the portfolio weights that is amplified in the case where there are non-zero primary balances and asymmetries in these balances. As may be seen in chart 9 below, at a particular level of correlation - (-0.5) in this example, there is a large change in the optimal portfolio. This swing from negative to positive values of α1, is seen to rise with increasing net primary balances. This effect arises from a change in sign of the denominator in equation (20) at the point of inversion. If reserves are managed at near this level major shifts in portfolio allocations are prescribed for small changes in the assumed correlations of the interest rate and exchange rate processes.
Currency Composition of Reserves Variation with Correlation (e1,r1)
Citation: IMF Working Papers 1991, 109; 10.5089/9781451946086.001.A001
VI. Reserve Level and Portfolio Choice-Implications for Intervention
The above analysis indicates the interaction of the level of reserves with the optimal portfolio choice. This consideration is an important one for the use of reserves in exchange market intervention. As reserves are drawn down the optimal allocation of reserves changes. Consequently, the currency composition of the transactions in exchange market intervention - as in all uses of reserves, will depend on the level of reserves. The composition of these transactions may, however, be inferred from the current reserve position and the net change required for the intervention, where the transaction with currency “i” is given by,
where A is the current level of reserves and B is the final level of reserves. These levels of reserves are those determined from the exchange rate policy of the country and are exogenous to this model. Substituting in the optimal portfolio proportion(1) and integrating we have,
and the portfolio weight for the transaction is simply ΔR1/ΔR where ΔR = B-A. Thus the transactions in each currency may be chosen to keep the optimal balance after the intervention. As indicated earlier, small economies are price takers and relative sizes in currency transactions will not affect bilateral rates significantly among the major currencies. However, additional adjustment of currency allocation will be necessary if the exchange rate relative to trading partners is changed by the process of intervention, as this will impact the imbalance in the primary account and the resulting optimal currency portfolio.
VII. Some General Remarks on Currency Diversification
A number of observations need to be made with regard to the general applicability of this approach. First, it should be stressed that the method described above is designed to reduce the risk faced by central banks in managing their foreign exchange reserves, and sovereign debt portfolio. The management of these assets and liabilities consists of identifying the stochastic properties (means, variances, and covariances) of relevant investment instruments as well as the major components of its international transactions. On the assumption that these parameters are fairly constant, a central bank or monetary authority can then compute the particular currency composition that minimizes the variance of its currency holdings or liabilities in real terms. Thus, no speculation on expected exchange rate changes is involved in this approach as in the classical portfolio model. (Although assumptions are made on the future volatility of exchange rates, interest rates and prices). Moreover, it explicitly adopts the assumption that the monetary authority wishes to minimize the risk of changes in foreign currency asset values. This simplifying--and perhaps extreme--assumption rules out other positions on the efficiency locus that would involve portfolios of higher expected return but also greater risk. This simplification allows treatment of the main contribution of this paper which is related to the effects of the primary account imbalance on the portfolio choice problem, without the complications introduced by mean-variance tradeoffs also addressed in classical portfolio theory. However, including mean-variance tradeoffs would be consistent with, and complementary to this model.
In the presence of transactions cost the active management of reserves using any portfolio approach including the one proposed, would incur an additional cost in changing the portfolio composition over time. However, judging from the level of active management of investment portfolios world wide, as compared to passive “buy and hold” strategies, it is likely that the former approach provides greater value for investors, and is thus likely to be more suited for the management of Central Bank reserves.
It should be noted that there are constraints on the extent to which monetary authorities can adjust the composition of their foreign currency reserves. Nearly all countries have exchange rate objectives, as mentioned above, and exchange market intervention designed to achieve these objectives may well cause departures in the actual currency composition of the portfolio from the composition which minimizes the variance of the portfolio. For G-7 countries exchange rate objectives are overriding and would appear to take precedence over portfolio considerations. The extremely large increase in the dollar reserve assets of Germany, Japan, and the United Kingdom in 1986-1988 is a vivid example of the extent to which exchange market intervention can affect the currency composition of a country’s reserves. It would be inconsistent with the exchange rate objectives of these countries to re-allocate the U.S. dollars acquired in support operations to investments in other currencies, as such dollar sales would tend to result in a depreciation of the dollar.
For most other countries, however, exchange market operations are unlikely to have significant effects on the exchange rates of the major industrial countries. Of course, the purchase or sale of a major reserve asset currency against the local currency will generate fluctuations in holdings of that reserve currency. Such intervention will typically lead to departure from that composition of reserves associated with a minimum--variance portfolio. However, to the extent that countries maintain working balances as well as other balances of exchange reserves that can be held for longer-term investment purposes, it will appear feasible to adjust over time the composition of foreign exchange reserves to approximate a minimum-variance portfolio. Alternatively the intervention in exchange markets may be conducted in a portfolio of currencies, to maintain the minimum variance. To the extent that countries have particular pegging arrangement--for example--to the U.S. dollar, sterling, French franc, or yen-- they would tend to hold more of these currencies in their reserves. Such pegging arrangements will be consistent with the approach suggested here.
VIII. Conclusions
An approach for the optimal diversification of central bank reserves (or sovereign debt) to minimize risk is proposed in a simple two period framework. It is assumed that the central bank decision on the net level of reserves is based on its preferences related to its intervention strategy, and is exogenous to this model. Therefore, the prescription of such an approach for portfolio adjustments would be subordinated to any policy related intervention in the exchange rate market. An approach such as this would be appropriate for small open economies that are price takers in that it will not appreciably alter bilateral rates among the major currencies in which investments are made. Besides specifying the optimal composition of the reserve portfolio, it may be inferred from the model that countries that have low net reserve levels relative to primary surpluses or deficits need to make larger transfers among currency allocations as reserve levels change, than countries with high levels of reserves. Further as the optimal allocation of reserves changes as reserves are drawn down or increased, the currency composition of the transactions in exchange market intervention will also depend on the magnitude of the intervention.
Finally, it needs to be emphasized that the general portfolio approach to reserve and debt diversification outlined here is a conservative strategy. It has as its overall objective the minimization of the variance of a country’s currency reserves or debt caused by exchange rate, interest rate and price changes. As such, it would appear consistent with the general principle of central banking operations in that the variation in aggregate wealth or purchasing power is minimized.
Appendix
Consider the optimal portfolio proportions as expressed equations [12] and [13]. These may be represented in terms of the variance covariance matrix of exchange and interest rates and those of the prices by making substitutions as follows.
Let Eoi = EiPEiteit, Ioi = IiPIiteit, Di = Dieit denote the export, import, and private debt service payments respectively in domestic currency terms. Further let λi = (Ei-Ii) represent the primary transactions balance excluding interest payments in terms of the domestic currency. It is readily seen that the country is in surplus when λi<0 and has a net deficit with the ‘i’th country when λi<0. The results in equation [12] and [13] may be represented in terms of the variances and covariances of the individual variables - the interest rates exchange rates and prices. Putting Vii, Vij for these Variances and Covariance respectively, where i and j represent the marginal returns on interest rates exchange rates and prices, the elements of the matrix V are given by;
substituting in [12] and [13] gives the optimal allocation.
References
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I would like to acknowledge helpful guidance from Peter Clark and valuable comments, and suggestions from numerous colleagues, especially Stijn Classens, Warren Coats, Orlando Roncesvalles, and Gary O’Callaghan. This paper was written while the author was in the Financial Relations Division of the Treasurer’s Department.
This simplification may be removed if controls on investment by foreigners is relaxed and monetary policy is endogenized.
This basic approach follows the analysis in Clark (1988).
Formal treatment of such longer term intervention would need a more complex model than the simple two period model in this paper.
However, the pricing of goods and services may “strategically” set and consequently the invoicing currency may not be the “true” underlying currency.
This objective is plausible for a central bank. The analysis could easily be extended to be conditional on a preferred expected return.
Although the analysis here does not assume any such constraints, the analysis in the next section may be extended to include such cases by using the Kuhn-Tucker inequality conditions that prevent negative investments. This analysis is excluded in the interest of simplicity. Such constraints would force investment into “second best” currencies when the non-negativity constraints become binding.
This model is similar but distinct from the usual investment portfolio problems in financial economics (see for example Sharpe [1981]). The solution finds the optimum portfolio for investments but includes consideration of the income from trade and payments.
The case of nominal reserves is subsumed in this solution by setting the price index to a constant.
There is some flexibility in the choice of deflator used to represent the changes in “purchasing power.”
Principal plus interest.
The value of the SDR is the value of 0.572 units of U.S. dollars, 0.453 units of deutsche marks, 31.8 units of yen, 0.80 units of french francs and 0.0812 units of the pound sterling effective January 1, 1991.
For example the standard deviation for daily exchange rates for the deutsche mark, French franc are 0.0006, and that for the pound sterling and the yen are 0.007 and 0.008 respectively during the last six months of 1990.
